Geometric Phase Shift in a Neural Oscillator

This paper studies the eeect of slow cyclic variation of parameters on the phase of the oscillating Morris-Lecar model. In chemical oscillators it is known that a phase shift, called the geometric phase shift, is observed upon return to an initial point in parameter space. We nd geometric phase shifts for a two-parameter variation in the Morris-Lecar model. As with the chemical oscillator, the magnitude of the shift is proportional to the area enclosed by the path traced through the parameter space. It is argued that the geometric phase shift may subserve many biological functions. We conclude that the geometric phase shift may be functionally relevant for neural computation. 1 Background Oscillations are found throughout the nervous system and are believed to play an important role in brain processing (Llinn as, 1988). Recent experimental ndings indicate that phase information is signiicant in neural computation involving oscillatory activity. For example, O'Keefe and Recce (1993) found rat hippocampal place cells ring at speciic phases of the oscillatory theta rhythm to signal the relative locations of environmental landmarks. Cells ring at an early phase signal the presence of landmarks ahead of the animal, whereas those ring at a late phase signal for landmarks located behind the animal. The utility of phase information is also apparent in other brain areas. In the gamma band, phase information may code for signal strength (Hoppeld, 1995) or serve as a binding code (Singer, 1995). Finding evidence of phase coding as an important factor in oscillatory neural processes suggests the potential signiicance in neurobiology of the geometric phase shift, a well-known phenomenon in Physics (Berry, 1988). When the parameters of an oscillating dynamical system are adiabat-ically (slowly) varied to trace a closed path in parameter space, a phase shift is observed upon This is a DRAFT VERSION of a paper we intend to expand. We welcome any comments or suggestions.